Free vibration of undamped Euler-Bernoulli and Timoshenko beams by using spectral finite element method for different boundary conditions

This paper describes spectral finite element (SFE) formulation and solution for free vibration of Euler-Bernoulli and Timoshenko beams. Formulation based on SFE method includes partial differential equations of motion, spectral displacement field, dynamic shape functions, and dynamic stiffness matrix. Dynamic shape functions in frequency-domain are derived from the exact solution of governing wave equations. By considering free vibration of Euler-Bernoulli and Timoshenko beams, natural frequencies for different boundary conditions (simply-supported, clamped and cantilever beams) are derived. In the SFE model, whole length of beam is represented by one or two spectral elements and in finite element (FE) model, total number of finite elements is varied in order to improve its solutions. The accuracy of results obtained from SFE formulation is compared with those of FE method and analytical formulations, whenever available. The SFE results display remarkable superiority compared to FE results in reducing number of elements as well as increasing numerical accuracy.

کلیدواژه ها:

spectral finite element ، Euler-Bernoulli beam ، Timoshenko beam ، dynamic stiffness matrix ، dynamic shape function ، natural frequency ، boundary condition